Sometimes there is confusion over the way in which characteristics like transitivity interact with the sub/super property hierarchy. As far as transitivity is concerned, the characteristic is not “inherited” by sub-properties — we cannot infer that the sub property of a transitive property is transitive. The same holds for super properties.

To illustrate this, consider the following example. We have three (object) properties: knows, hasFriend, marriedTo. One of these (hasFriend) is transitive (now you might question this as a piece of modelling, but please just go with my rosy world-view that all the friends of my friends are also friends), and the properties are arranged in a hierarchy. In Manchester syntax we would have:

If we look at the ontology presented above, we can see that all the axioms hold — the subproperty axioms are being respected, as is the transitivity of hasFriend. Thus this situation is a model of the ontology.

Now if we consider isMarriedTo, we can see that our conditions for transitivity do not hold. There are three elements with isMarriedTo(Betty,Charlie) and isMarriedTo(Charlie,Daphne), but we do not have isMarriedTo(Betty,Daphne). So we cannot infer that isMarriedTo is transitive from the axioms. Similarly, there are three elements where knows(Arthur,Betty) and knows(Betty,Charlie) but we don’t have knows(Arthur,Charlie).

Recall that the inferences we can make from an ontology or collection of axioms are those things that necessarily hold in all models of the ontology. This little sample model provides us a “witness” for the fact that we cannot infer that knows is transitive from the axioms. Similarly, we cannot infer that isMarriedTo is transitive.

Of course, this is just saying that we can’t in general make such an inference. We are not saying that superproperties cannot (sometimes) be transitive. If we add to our interpretation the fact that Arthur knows Charlie and Daphne, then in this interpretation, knows is indeed transitive. And if we allow Betty to marry Daphne — hey, it’s 2013! — then we have a transitive subproperty (in this interpretation).

On the topic of transitivity and sub properties, the thesaurus representation SKOS uses a common modelling pattern, where a non-transitive property (skos:broader) has a transitive superproperty (skos:broaderTransitive) defined. The superproperty is not intended to be used for asserting relationships, but can be used to query for transitive chains of skos:broader relationships (assuming our query engine is performing inference). As we now know, this doesn’t mean that skos:broader is necessarily transitive.

This pattern is also often used for representing partonomy. Here we would use a (non-transitive) hasDirectPart for asserting parts of a whole, with a transitive superproperty hasPart allowing us to query the transitive closure. We can use counting with hasDirectPart — for example min or max cardinality restrictions — which we would not be able to do in OWL DL if hasDirectPart was transitive, due to restrictions relating to simple properties (see the OWL2 Structural Specification).

For other property characteristics the situation is different. For example, a subproperty of a functional property must be functional. Why? We’ll leave that as an exercise for the interested reader……